To disable the screen wake when an SMS is received on your Galaxy S4, follow my guide below.
You will need to download 2 apps from Play Store to accompolish. It's called Ringo Lite and Light Manager. Install Ringo Lite and use it for your notification sound, vibrate and such. You can also use this to set individual contacts to silent or other tones. Next app is Light Manager for your LED light.
Next, disable sounds for SMS or simply set to silent. Now go to stock messaging app, settings and untick notifications. Now set your Light Manager app to control the color of your LED indicator.
You no longer will have a screen wake when an SMS is received. You will have sound and LED set to your color only.
I tried this, but even though I got the wake up screen to stop when I received a text, I no longer (and I tried and tried!!) was getting a flashing light to notify me of anything...not texts, not emails, not missed calls, not voicemail...!! So I started from scratch again and uninstalled both apps.
Then I went looking for another app and I found SMS Popup (I do not use the Popup feature) and it works absolutely perfectly!! No more wake up screen when I get a text. I can choose what color and frequency of notification flash for everything. My phone vibrates, plays a sound (if I choose it to, depending on the time of day), flashes a light.... everything is perfect. All I needed was that one app.
This was my method at first. I have since found an easier method. Download Textra. It's a new messaging app just like Handscent and GoSMS, but this one has way more features. It even has built in disable screen wake, different sounds for each contact and dark mode for battery. It's an all around perfect messaging app. Thanks for your solution as well. The more we have, the more options people have to get rid of that screen wake annoyance.